More precisely, we find the values of \((n_{E},n_{E^{\prime }})\)  for the rooted
model to change in the sequence \((0,0)\) , \((2,0)\) , \((2,2)\) , \((2,4)\) ,
\((4,4)\) , \((6,4)\) , \((6,6)\)  for \(E=0,2\pi /3\)  (\(E=\pi /4,3\pi /4\) )
and \(E^{\prime }=\pi /3,\pi \)  (\(E^{\prime }=0,\pi /2,\pi \) ) in Fig. REF (b)
(Fig. REF (d)) with the increase of \(J_{1}\) , while
the winding numbers of the parent model change as \((\nu _{0},\nu _{\pi })=(0,0)\) ,
\((1,0)\) , \((1,-1)\) , \((1,-2)\) , \((2,-2)\) , \((3,-2)\) , \((3,-3)\)  during
the process [1]}, confirming the relations in Eqs. (REF )
and (REF ).
In Fig. REF , we present detailed examples of degenerate edge modes at fractional quasienergies and bulk
skin modes that coexist with these topological edge states in the systems described by \(U_{1/q}\)  for \(q=2,3,4\) .
The numbers of edge modes found there coincide with our theoretical predictions.
All these results reveal the power of
our strategy in constructing \(q\) th-root FTIs
for any \(q\in \mathbb {Z}^+\) . In the following subsection, we will demonstrate
the applicability of the same routine in the construction of \(q\) th-root FSOTIs.
